We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős–Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if, after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size, as well as the fraction of vertices contained in the giant color-avoidin g connected component. It is known that these quantities converge, and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the fraction of vertices contained in the giant color-avoiding connected component, and we give a simpler asymptotic expression for it in the barely supercritical regime. In addition, in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.

A partition $\lambda $ of n is said to be nearly self-conjugate if the Ferrers graph of $\lambda $ and its transpose have exactly $n-1$ cells in common. The generating function of the number of such partitions was first conjectured by Campbell and recently confirmed by Campbell and Chern (‘Nearly self-conjugate integer partitions’, submitted for publication). We present a simple and direct analytic proof and a combinatorial proof of an equivalent statement.

By combining the generating function approach with the Lagrange expansion formula, we evaluate, in closed form, two multiple alternating sums of binomial coefficients, which can be regarded as alternating counterparts of the circular sum evaluation discovered by Carlitz [‘The characteristic polynomial of a certain matrix of binomial coefficients’, Fibonacci Quart. 3(2) (1965), 81–89].

For a graph $\Gamma$, let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$. We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$, we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

This paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

Within the online media universe, there are many underlying communities. These may be defined, for example, through politics, location, health, occupation, extracurricular interests or retail habits. Government departments, charities and commercial organisations can benefit greatly from insights about the structure of these communities; the move to customer-centred practices requires knowledge of the customer base. Motivated by this issue, we address the fundamental question of whether a sub-network looks like a collection of individuals who have effectively been picked at random from the whole, or instead forms a distinctive community with a new, discernible structure. In the former case, to spread a message to the intended user base it may be best to use traditional broadcast media (TV, billboard), whereas in the latter case a more targeted approach could be more effective. In this work, we therefore formalise a concept of testing for sub-structure and apply it to social interaction data. First, we develop a statistical test to determine whether a given sub-network (induced sub-graph) is likely to have been generated by sampling nodes from the full network uniformly at random. This tackles an interesting inverse alternative to the more widely studied “forward” problem. We then apply the test to a Twitter reciprocated mentions network where a range of brand name based sub-networks are created via tweet content. We correlate the computed results against the independent views of 16 digital marketing professionals. We conclude that there is great potential for social media based analytics to quantify, compare and interpret online brand allegiances systematically, in real time and at large scale.

In this paper we give an extension of the results of the generalized waiting time problem given by El-Desouky and Hussen (1990). An urn contains m types of balls of unequal numbers, and balls are drawn with replacement until first duplication. In the case of finite memory of order k, let ni be the number of type i, i = 1, 2, …, m. The probability of success pi = ni/N, i = 1, 2, …, m, where ni is a positive integer and Let Ym,k be the number of drawings required until first duplication. We obtain some new expressions of the probability function, in terms of Stirling numbers, symmetric polynomials, and generalized harmonic numbers. Moreover, some special cases are investigated. Finally, some important new combinatorial identities are obtained.

The probability h(n, m) that the block counting process of the Bolthausen-Sznitman n-coalescent ever visits the state m is analyzed. It is shown that the asymptotic hitting probabilities h(m) = limn→∞h(n, m), m ∈ N, exist and an integral formula for h(m) is provided. The proof is based on generating functions and exploits a certain convolution property of the Bolthausen-Sznitman coalescent. It follows that h(m) ∼ 1/log m as m → ∞. An application to linear recursions is indicated.

Combinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free.

The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating function G and a maximum principle which, we prove, is satisfied by every fixed point of G. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’, Util. Math. 88 (2012), 223–235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight $n$. He also proved that the generating function for $f(n), $ the number of 1-shell totally symmetric plane partitions of weight $n$, is given by

$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$

$f(n)$

Let (Sk)k≥1 be the classical Bernoulli random walk on the integer line with jump parametersp ∈ (0,1) andq = 1 − p. The probability distribution of the sojourntime of the walk in the set of non-negative integers up to a fixed time is well-known, butits expression is not simple. By modifying slightly this sojourn time through a particularcounting process of the zeros of the walk as done by Chung & Feller [Proc.Nat. Acad. Sci. USA 35 (1949) 605–608], simpler representationsmay be obtained for its probability distribution. In the aforementioned article, only thesymmetric case (p = q = 1/2) isconsidered. This is the discrete counterpart to the famous Paul Lévy’s arcsine law forBrownian motion.

In the present paper, we write out a representation for this probabilitydistribution in the general case together with others related to the random walk subjectto a possible conditioning. The main tool is the use of generating functions.

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times ln between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of ln / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

A computer aided method using symbolic computations that enables the calculation of thesource terms (Boltzmann) in Grad’s method of moments is presented. The method is extremelypowerful, easy to program and allows the derivation of balance equations to very highmoments (limited only by computer resources). For sake of demonstration the method isapplied to a simple case: the one-dimensional stationary granular gas under gravity. Themethod should find applications in the field of rarefied gases, as well. Questions ofconvergence, closure are beyond the scope of this article.

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.

Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process dj ≤ B balls are placed in distinct urns with uniform probability. Let Mi(j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M0(j) and M1(j) is obtained. A multivariate generating function for the joint factorial moments of Mi(j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the dj, j ≥ 1, are independent, identically distributed random variables is investigated.